This document provides key concepts and formulas regarding three-dimensional geometry. It defines formulas for finding the distance between two points, dividing a line segment between two points internally or externally, finding direction ratios and direction cosines of a line, vector and Cartesian equations of a line, finding the angle between two lines or planes, determining if two lines or planes are perpendicular or parallel, finding the distance of a point from a plane, determining if two lines are coplanar, and finding the angle between a line and plane. Examples of each type of concept and formula are provided.

1. XII – Maths 111
CHAPTER 11
THREE DIMENSIONAL GEOMETRY
POINTS TO REMEMBER
 Distance between points P(x1, y1, z1) and Q(x2, y2, z2) is
     2 2 2
2 1 2 1 2 1 .x x y y z zPQ     

 (i) The coordinates of point R which divides line segment PQ where
P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally are
2 1 2 1 2 1 ., ,
mx nx my ny mz nz
m n m n m n
   
   
(ii) The co-ordinates of a point which divides join of (x1, y1, z1) and
(x2, y2, z2) in the ratio of m : n externally are
2 1 2 1 2 1 ., ,
mx nx my ny mz nz
m n m n m n
   
   
 Direction ratios of a line through (x1, y1, z1) and (x2, y2, z2) are x2 – x1,
y2 – y1, z2 – z1.
 Direction cosines of a line whose direction ratios are a, b, c are given by
2 2 2 2 2 2 2 2 2
, , .
a b c
l m n
a b c a b c a b c
     
     
 (i) Vector equation of a line through point a

and parallel to vector
b

is .r a b  
  
(ii) Cartesian equation of a line through point (x1, y1, z1) and having
direction ratios proportional to a, b, c is

2. 112 XII – Maths
1 1 1
.
x x y y z z
a b c
  
 
 (i) Vector equation of line through two points
 and is .a b r a b a   
     
(ii) Cartesian equation of a line through two points (x1, y1, z1) and
(x2, y2, z2) is
1 1 1
2 1 2 1 2 1
.
x x y y z z
x x y y z z
  
 
  
 Angle ‘’ between lines     
     
1 1 2 2andr a b r a µ b is given
by 1 2
1 2
cos .
b b
b b

 
 
 
 Angle  between lines 1 1 1
1 1 1
x x y y z z
a b c
  
  and 2
2
x x
a


2 2
2 2
y y z z
b c
 
 is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
cos .
a a b b c c
a b c a b c
 
 
   
 Two lines are perpendicular to each other if
1 2 0b b 
 
or a1a2 + b1b2 + c1c2 = 0.
 Equation of plane :
(i) At a distance of p unit from origin and perpendicular to n is
r n p 

and corresponding Cartesian form is lx + my + nz = p
when l, m and n are d.c.s of normal to plane.
(ii) Passing through a

and normal to

n is  . 0
 
nr a and
corresponding Cartesian form is a(x – x1) + b (y – y1) +
c(z – z1) = 0 where a, b, c are d.r.’s of normal to plane and
(x1, y1, z1) lies on the plane.
(iii) Passing through three non collinear points is
      0r a b a c a      
     

3. XII – Maths 113
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
or 0
x x y y z z
x x y y z z
x x y y z z
  
   
  
(iv) H a vin g in te rce p ts a, b and c on co-ordinate axis is
1
x y z
a b c
   .
(v) Planes passing through the line of intersection of planes
1 1r n d 
 
and 2 2r n d 
 
is
   1 1 2 2 0.r n d r n d     
   
 (i) Angle ‘’ between planes 1 1 2 2andr n d r n d   
   
is
given by 1 2
1 2
cos .
n n
n n

 
 
 
(ii) Angle  between a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2
is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
cos .
a a b b c c
a b c a b c
 
 
   
(iii) Two planes are perpendicular to each other iff 
 
1 2. 0n n or
a1a2 + b1b2 + c1c2 = 0.
(iv) Two planes are parallel iff 1 2 for some scalern n 
 
1 1 1
2 2 2
0 or
a b c
a b c
    .
 (i) Distance of a point  

a from plane   isr n d 
 
.
 
 

a n d
n
(ii) Distance of a point (x1, y1, z1) from plane ax + by + cz = d is
1 1 1
2 2 2
.
ax by cz d
a b c
  
 
12. (i) Two lines 1 1 2 2andr a b r a b     
     
are coplanar.
Iff  2 1a a 
 
 1 2 0b b 
 
and equation of plane,
containing these lines is    1 1 2 0.r a b b  
   

4. 114 XII – Maths
(ii) Two lines 1 1 1
1 1 1
and
x x y y z z
a b c
  
 
2
2
x x
a

 2 2
2 2
y y z z
b c
 
 are coplanar Iff
2 1 2 1 2 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
  

and equation of plane containing them is
1 1 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
  

.
 (i) The angle  between line r a b  
  
and plane r n d 
 
is given as sin .
b n
b n

 
 
 
90°–
b

(ii) The angle  between line
1 1 1
1 1 1
x x y y z z
a b c
  
  and plane
a2x + b2y + c2 z = d is given as
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
sin .
a a b b c c
a b c a b c
 
 
   
(iii) A line r a b  
  
is parallel to plane .r n d
 
0b n  
 
or a1a2 + b1b2 + c1c2 = 0.
r = a + b
r . n = d

5. XII – Maths 115
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1. What is the distance of point (a, b, c) from x-axis?
2. What is the angle between the lines 2x = 3y = – z and 6x = – y = – 4z?
3. Write the equation of a line passing through (2, –3, 5) and parallel to line
1 2 1
.
3 4 1
x y z  
 

4. Write the equation of a line through (1, 2, 3) and perpendicular to
  5.3r i j k  

 
5. What is the value of  for which the lines
1 3 1
and
2 5
  
 

x y z
2 1
3 2 2
 
 

x y z
are perpendicular to each other.
6. If a line makes angle , , and  with co-ordinate axes, then what is the
value of
sin2  + sin2  + sin2?
7. Write line    2r i j j k   

   into Cartesian form.
8. If the direction ratios of a line are 1, –2, 2 then what are the direction
cosines of the line?
9. Find the angle between the planes 2x – 3y + 6z = 9 and xy – plane.
10. Write equation of a line passing through (0, 1, 2) and equally inclined to
co-ordinate axes.
11. What is the perpendicular distance of plane 2x – y + 3z = 10 from origin?
12. What is the y-intercept of the plane x – 5y + 7z = 10?
13. What is the distance between the planes 2x + 2y – z + 2 = 0 and
4x + 4y – 2z + 5 = 0.
14. What is the equation of the plane which cuts off equal intercepts of unit
length on the coordinate axes.
15. Are the planes x + y – 2z + 4 = 0 and 3x + 3y – 6z + 5 = 0 intersecting?
16. What is the equation of the plane through the point (1, 4, – 2) and parallel
to the plane – 2x + y – 3z = 7?

6. 116 XII – Maths
17. Write the vector equation of the plane which is at a distance of 8 units
from the origin and is normal to the vector  2 2 .i j k  
18. What is equation of the plane if the foot of perpendicular from origin to
this plane is (2, 3, 4)?
19. Find the angles between the planes   1 and2 2r i j k  

  
  0.3 6 2r i j k  

  
20. What is the angle between the line
1 2 1 2 –
3 4 –4
x y z 
  and the
plane 2x + y – 2z + 4 = 0?
21. If O is origin OP = 3 with direction ratios proportional to –1, 2, – 2 then
what are the coordinates of P?
22. What is the distance between the line   2 – 2 3 4r i j k i j k     

   
from the plane  . – 5 – 5 0.r i j k  

 
23. Write the line 2x = 3y = 4z in vector form.
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
24. The line
4 2 4
1 2 2
x y k z  
 

lies exactly in the plane
2x – 4y + z = 7. Find the value of k.
25. Find the equation of a plane containing the points (0, –1, –1), (–4, 4, 4)
and (4, 5, 1). Also show that (3, 9, 4) lies on that plane.
26. Find the equation of the plane which is perpendicular to the plane
  8 05 3 6r i j k   

  & which is containing the line of intersection
of the planes   42 3r i j k  

  and   5 0.2r i j k   

 
27. If l1, m1, n1, and l2, m2, n2 are direction cosines of two mutually
perpendicular lines, show that the direction cosines of line perpendicular
to both of them are
m1n2 – n1m2, n1l2 – l1n2, l1m2 – m1l2.

7. XII – Maths 117
28. Find vector and Cartesian equation of a line passing through a point with
position vectors 2i j k   and which is parallel to the line joining the
points with position vectors – 4i j k   and 2 2 .i j k  
29. Find the equation of the plane passing through the point (3, 4, 2) and
(7, 0, 6) and is perpendicular to the plane 2x – 5y = 15.
30. Find equation of plane through line of intersection of planes
  12 02 6r i j  

  and   03 4r i j k  

  which is at a unit
distance from origin.
31. Find the image of the point (3, –2, 1) in the plane 3x – y + 4z = 2.
32. Find the equation of a line passing through (2, 0, 5) and which is parallel
to line 6x – 2 = 3y + 1 = 2z – 2.
33. Find image (reflection) of the point (7, 4, – 3) in the line
– 1 2
.
1 2 3
x y z 
 
34. Find equations of a plane passing through the points (2, –1, 0) and
(3, –4, 5) and parallel to the line 2x = 3y = 4z.
35. Find distance of the point (– 1, – 5, – 10) from the point of intersection
of line
  
 
2 1 2
3 4 12
x y z
and the plane x – y + z = 5.
36. Find equation of the plane passing through the points (2, 3, – 4) and
(1, –1, 3) and parallel to the x–axis.
37. Find the distance of the point (1, –2, 3) from the plane x – y + z = 5,
measured parallel to the line  

.
2 3 6
x y z
38. Find the equation of the plane passing through the intersection of two
plane 3x – 4y + 5z = 10, 2x + 2y – 3z = 4 and parallel to the line
x = 2y = 3z.
39. Find the distance between the planes 2x + 3y – 4z + 5 = 0 and
 . 4 6 – 8 11.r i j k 

  
40. Find the equations of the planes parallel to the plane x – 2y + 2z – 3 = 0
whose perpendicular distance from the point (1, 2, 3) is 1 unit.

8. 118 XII – Maths
41. Show that the lines
1 3 5
and
3 5 7
x y z  
 
2
1
x 

4
3
y 

6
5
z 
intersect each other. Find the point of
intersection.
42. Find the shortest distance between the lines
2 3r l j k   

    2 3 4i j k     and
   .2 4 5 3 4 5r i j k i j k     

   
43. Find the distance of the point (–2, 3, –4) from the line
2
3


x
2 3 3 4
4 5
 

y z
measured parallel to the plane 4x + 12y – 3z + 1 = 0.
44. Find the equation of plane passing through the point (–1, –1, 2) and
perpendicular to each of the plane
   2 and 6.2 3 3 5 4r ri j k i j k      
 
   
45. Find the equation of a plane passing through (–1, 3, 2) and parallel to
each of the line
1 2 3
x y z
  and
2 1 1
.
–3 2 5
x y z  
 
46. Show that the plane   73 5r i j k  

  contains the line
   .3 3 3r i j k i j    

   
LONG ANSWER TYPE QUESTIONS (6 MARKS)
47. Check the coplanarity of lines
   –3 5 –3 5r i j k i j k      

   
   – 2 5 – 2 5r i j k µ i j k     

   
If they are coplanar, find equation of the plane containing the lines.
48. Find shortest distance between the lines :
8 19 10 15 29 5
and
3 16 7 3 8 5
x y z x y z     
   
 
.

9. XII – Maths 119
49. Find shortest distance between the lines :
      1 2 3 2r i j k       

 
      .1 2 1 2 1r i j k       

 
50. A variable plane is at a constant distance 3p from the origin and meets
the coordinate axes in A, B and C. If the centroid of ABC is (), then
show that –2 + –2 + –2 = p–2.
51. A vector n

of magnitude 8 units is inclined to x–axis at 45°, y axis at
60° and an acute angle with z-axis. If a plane passes through a point
 2, –1,1 and is normal to n

, find its equation in vector form.
52. Find the foot of perpendicular from the point 2 5i j k   on the line
 11 2 8r i j k  

   10 4 11i j k     . Also find the length of the
perpendicular.
53. A line makes angles , , ,  with the four diagonals of a cube. Prove
that
2 2 2 2 4
cos cos cos cos .
3
       
54. Find the equation of the plane passing through the intersection of planes
2x + 3y – z = – 1 and x + y – 2z + 3 = 0 and perpendicular to the plane
3x – y – 2z = 4. Also find the inclination of this plane with xy-plane.
ANSWERS
1. 2 2
b c 2. 90°
3.
2 3 5
.
3 4 1
x y z  
 

4.    2 3 3r i j k i j k      

   
5.  = 2 6. 2
7.
1 1
.
0 2 1
x y z 
 

8.
1 2 2
, ,
3 3 3
 

10. 120 XII – Maths
9. cos–1 (6/7).
10.  
1 2
, 0
x y z
a R
a a a
 
   
11.
10
14
12. –2
13.
1
6
14. x + y + z = 1
15. No 16. –2x + y – 3z = 8
17.  2 2 24r i j k   

  18. 2x + 3y + 4z = 29
19.
1 11
cos
21
  
  20. 0 (line is parallel to plane)
21. (–1, 2, –2) 22.
10
3 3
23.  6 4 3r o i j k    
 
  
24. k = 7 25. 5x – 7y + 11z + 4 = 0.
26.  –51 – 15 50 173r i j k  

 
28.    2 – 2 – 2r i j k i j k    

    and
2 1 1
.
2 2 1
x y z  
 

29. x – 2y + 3z = 1
30.    8 4 8 12 0 or 4 8 8 12 0r i j k r i j k          
 
   
31. (0, –1, –3) 32.
2 5
.
1 2 3
x y z 
 
33.
47 18 43
, ,
7 7 7
 
  34. 29x – 27y – 22z = 85
35. 13 36. 7y + 4z = 5

11. XII – Maths 121
37. 1 38. x – 20y + 27z = 14
39.
21
units.
2 29
40. x – 2y + 2z = 0 or x – 2y + 2z = 6
41.
1 1 3
, – , –
2 2 2
 
   42.
1
6
43.
17
2
44.  9 17 23 20r i j k   

 
45. 2x – 7y + 4z + 15 = 0
47. x – 2y + z = 0 48.
16
7
49.
8
29
51.  2 2r i j k   

 
52. 2 3 , 14i j k  
54.
1 4
7 13 4 9, cos
234
x y z
  
     
.